R. Álvarez-Nodarse

On the q -polynomials: a distributed study

In this paper we present a uni1ed distributional study of the classical discrete q-polynomials (in the Hahn’s sense). From the distributional q-Pearson equation we will deduce many of their properties such as the three-term recurrence relations, structure relations, etc. Also several characterizations of such q-polynomials are presented. c

On the “Favard theorem” and its extensions

In this paper we present a survey on the “Favard theorem” and its extensions. c 2001 Elsevier Science B.V. All rights reserved.

Asymptotic properties of generalized Laguerre orthogonal polynomials

Abstract In the present paper we deal with the polynomials L n ( α , M , N ) ( x ) orthogonal with respect to the Sobolev inner product (p,q) = 1 Г(α+1) ∫ 0 ∞ p(x)q(x)x α e −x dx + Mp(O)q(O) + Np′(O)q′(O), N, M ≥ O, α > −I , firstly introduced by Koekoek and Meijer in 1993 and extensively studied in the last years. We present some new asymptotic properties of these polynomials and also a limit relation between the zeros of these polynomials and the zeros of Bessel function J α ( x ). The results are illustrated with numerical examples. Also, some general asymptotic formulas for generalizations of these polynomials are conjectured.

Modified Clebsch-Gordan-type expansions for products of discrete hypergeometric polynomials

Abstract Starting from the second-order difference hypergeometric equation satisfied by the set of discrete orthogonal polynomials ∗p n ∗ , we find the analytical expressions of the expansion coefficients of any polynomial rm(x) and of the product rm(x)qj(x) in series of the set ∗p n ∗ . These coefficients are given in terms of the polynomial coefficients of the second-order difference equations satisfied by the involved discrete hypergeometric polynomials. Here qj(x) denotes an arbitrary discrete hypergeometric polynomial of degree j. The particular cases in which ∗r m ∗ corresponds to the non-orthogonal families ∗x m ∗ , the rising factorials or Pochhammer polynomials ∗(x) m ∗ and the falling factorial or Stirling polynomials ∗x [m] ∗ are considered in detail. The connection problem between discrete hypergeometric polynomials, which here corresponds to the product case with m = 0, is also studied and its complete solution for all the classical discrete orthogonal hypergeometric (CDOH) polynomials is given. Also, the inversion problems of CDOH polynomials associated to the three aforementioned nonorthogonal families are solved.

q -Classical polynomials and the q -Askey and Nikiforov-Uvarov tableaus

Abstract In this paper we continue the study of the q -classical (discrete) polynomials (in the Hahns sense) started in Medem et al. (this issue, Comput. Appl. Math. 135 (2001) 157–196). Here we will compare our scheme with the well known q -Askey scheme and the Nikiforov–Uvarov tableau. Also, new families of q -polynomials are introduced.

WKB Approximation and Krall-Type Orthogonal Polynomials

We give a unified approach to the Krall-type polynomials orthogonal withrespect to a positive measure consisting of an absolutely continuous one‘perturbed’ by the addition of one or more Dirac deltafunctions. Some examples studied by different authors are considered from aunique point of view. Also some properties of the Krall-type polynomials arestudied. The three-term recurrence relation is calculated explicitly, aswell as some asymptotic formulas. With special emphasis will be consideredthe second order differential equations that such polynomials satisfy. Theyallow us to obtain the central moments and the WKB approximation of thedistribution of zeros. Some examples coming from quadratic polynomialmappings and tridiagonal periodic matrices are also studied.

Recurrence relations for connection coefficients between Q-orthogonal polynomials of discrete variables in the non-uniform lattice x(s)=q2S

We obtain the structure relations for q-orthogonal polynomials in the exponential lattice and from these we construct the recurrence relation for the connection coefficients between two families of polynomials belonging to the classical class of discrete q-orthogonal polynomials. An explicit example is also given.

Recurrence relations for radial wavefunctions for the Nth-dimensional oscillators and hydrogenlike atoms

We present a general procedure for finding recurrence relations of the radial wavefunctions for Nth-dimensional isotropic harmonic oscillators and hydrogenlike atoms.

Jacobi-Sobolev-type orthogonal polynomials: second-order differential equation and zeros

Abstract We obtain an explicit expression for the Sobolev-type orthogonal polynomials {Qn} associated with the inner product 〈p,q〉= ∫ −1 1 p(x)q(x)p(x) dx + A 1 p(1)q(1) + B 1 p(−1)q(−1) + A 2 p′(1)q′(1) + B 2 p′(−1)q′(−1) , where p(x) = (1 − x)α(1 + x)β is the Jacobi weight function, α,β> − 1, A1,B1,A2,B2⩾0 and p, q ∈ P, the linear space of polynomials with real coefficients. The hypergeometric representation (6F5) and the second-order linear differential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in [−1, 1] is studied. Furthermore, we obtain some estimates for the largest zero of Qn(x). Such a zero is located outside the interval [−1, 1]. We deduce his dependence of the masses. Finally, the WKB analysis for the distribution of zeros is presented.

On the q-polynomials in the exponential lattice x(s)=c 1 qs +c 3

The main goal of this paper is to continue the sutudy of the q-polynomials on non-uniform lattices by using the approach introduced by Nikiforov and Uvarov in 1983. We consider the q-polynomials on the non-uniform exponential lattice x(s)= c 1 qs +c 3 and study some of their properties (differentiation formulas, structure relations, represntation in terms of hypergeometric and basic hypergeometric functions, etc). Special emphasis is given to q-analogues of the Charlier orthogonal polynomials. For these polynomials (Charlier) we compute the main data, i.e., the coefficients of the three-term recurrence relation, structure relation, the square of the norm, etc, in the exponential lattices , respectively.